Identity Element for Power Structure

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Theorem

Let $\struct {S, \circ}$ be a magma whose underlying set $S$ is non-empty.

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.


Then:

a subset $J$ of $S$ is an identity element of $\struct {\powerset S, \circ_\PP}$

if and only if:

there exists an identity element $e$ of $\struct {S, \circ}$, such that $J = \set e$.


Proof

Sufficient Condition

Let $J \subseteq S$ such that $J$ is an identity element of $\struct {\powerset S, \circ_\PP}$.

We have:

\(\ds \forall A \in \powerset S: \, \) \(\ds J \circ_\PP A\) \(=\) \(\ds A\) Definition of Identity Element
\(\ds \leadsto \ \ \) \(\ds \set {j \circ a: j \in J, a \in A}\) \(=\) \(\ds A\) Definition of Operation Induced on Power Set
\(\ds \leadsto \ \ \) \(\ds \forall a \in A: \forall j \in J: \, \) \(\ds j \circ a\) \(=\) \(\ds a\)

That is, the elements of $J$ are all left identities for $\circ$.


Similarly:

\(\ds \forall A \in \powerset S: \, \) \(\ds A \circ_\PP J\) \(=\) \(\ds A\) Definition of Identity Element
\(\ds \leadsto \ \ \) \(\ds \set {a \circ j: j \in J, a \in A}\) \(=\) \(\ds A\) Definition of Operation Induced on Power Set
\(\ds \leadsto \ \ \) \(\ds \forall a \in A: \forall j \in J: \, \) \(\ds a \circ j\) \(=\) \(\ds a\)

That is, the elements of $J$ are all right identities for $\circ$.


So we have established that $J$ contains at least one element which is both a left identity and a right identity.

From Left and Right Identity are the Same, there is only one element of $J$, and it is the necessarily unique identity element for $\circ$.

That is:

there exists an identity element $e$ of $\struct {S, \circ}$, such that $J = \set e$.

$\Box$


Necessary Condition

Let $\struct {S, \circ}$ have an identity element $e$.

Let $J = \set e$.

Then we have:

\(\ds \forall A \in \powerset S: \, \) \(\ds A \circ_\PP J\) \(=\) \(\ds \set {a \circ j: j \in J, a \in A}\) Definition of Operation Induced on Power Set
\(\ds \) \(=\) \(\ds \set {a \circ e: a \in A}\) Definition of $J$
\(\ds \) \(=\) \(\ds \set {a: a \in A}\) Definition of Identity Element
\(\ds \) \(=\) \(\ds A\)

and similarly:

\(\ds \forall A \in \powerset S: \, \) \(\ds J \circ_\PP A\) \(=\) \(\ds \set {j \circ a: j \in J, a \in A}\) Definition of Operation Induced on Power Set
\(\ds \) \(=\) \(\ds \set {e \circ a: a \in A}\) Definition of $J$
\(\ds \) \(=\) \(\ds \set {a: a \in A}\) Definition of Identity Element
\(\ds \) \(=\) \(\ds A\)

So:

$\forall A \in \powerset S: A \circ_\PP J = A = J \circ_\PP A$

and it is seen that $J$ is an identity element for $\circ_\PP$.

$\blacksquare$


Sources

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